Finite Group Databasebeta

← #36365637

Subgroups — #36365637

Order 22144.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012199C211yesmaximal, characteristic, complemented
2#26012129C211yesmaximal, characteristic
3#26011852C211yesmaximal, characteristic, complemented
4#26011825C211yesmaximal, characteristic, complemented
5#26011805C211yesmaximal, characteristic, complemented
6#26011769C211yesmaximal, characteristic, complemented
7#26011739C211yesmaximal, characteristic, complemented
8#26011729C211yesmaximal, characteristic, complemented
9#26011715C211yesmaximal, characteristic, complemented
10#26011640C211yesmaximal, characteristic, complemented
11#26011592C211yesmaximal, characteristic, complemented
12#26011565C211yesmaximal, characteristic, complemented
13#26011492C211yesmaximal, characteristic, complemented
14#102×D173C211yesmaximal, characteristic, complemented
15#452×C173C211yesmaximal, characteristic, complemented
16C2×D4×D173C2211yescharacteristic
17C2×#14583813C2211yescharacteristic
18C2×#14583810C2211yescharacteristic, complemented
19C2×C8×D173C2211yescharacteristic
20#19149678C2211yescharacteristic, complemented
21#19149669—21no—
22#19149668C2211yescharacteristic
23#19149662—21no—
24#19149659—21no—
25#19149656—21no—
26#19149655C2211yescharacteristic
27#19149655C2211yescharacteristic
28#19149653C2211yescharacteristic
29#19149651—21no—
30#19149647—21no—
31Q8×Dic173C2211yescharacteristic, complemented
32D4×Dic173C2211yescharacteristic
33C4×C22⋊D173C2211yescharacteristic
34#19149639—21no—
35C2×C4⋊Dic173C2211yescharacteristic, complemented
36C4⋊C4×D173C2211yescharacteristic
37C2×#14583803—21no—
38#19149613C2211yescharacteristic, complemented
39#19149613C2211yescharacteristic, complemented
40#19149612C2211yescharacteristic, complemented
41#19149608C2211yescharacteristic, complemented
42#19149603C2211yescharacteristic, complemented
43#19149603C2211yescharacteristic, complemented
44#19149603C2211yescharacteristic, complemented
45#19149601C2211yescharacteristic
46#19149601C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49#19149599C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149589C2211yescharacteristic, complemented
52C2×#41×C173C2211yescharacteristic
53C2×Q16×C173C2211yescharacteristic, complemented
54C2×SD16×C173C2211yescharacteristic
55C22⋊Q8×C173C2211yescharacteristic, complemented
56#103×C173C2211yescharacteristic, complemented
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60C2×Dic346—21no—
61C2×D692C2311yescharacteristic
62C2×C22⋊D173—21no—
63C2×C22⋊D173—21no—
64C2×C22⋊D173D411yescharacteristic
65D4×D173—22no—
66D4×D173—41no—
67C22×Dic173—21no—
68C22×Dic173—21no—
69C2×C4×D173—21no—
70C2×C4×D173C2311yescharacteristic
71#14583813—41no—
72#14583810—41no—
73C2×#13063436C2311yescharacteristic
74C2×#13063436C2311yescharacteristic
75C8×D173—41no—
76C4⋊Dic173—21no—
77C4⋊Dic173—21no—
78C4⋊Dic173C2311yescharacteristic
79C4⋊Dic173C2311yescharacteristic
80C4⋊Dic173C2311yescharacteristic
81#14583805—21no—
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583805D411yescharacteristic
86C22⋊Dic173—21no—
87C22⋊Dic173D411yescharacteristic
88#14583803—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803D411yescharacteristic
94C4×Dic173—21no—
95C4×Dic173C2311yescharacteristic
96#41×C173—22no—
97#41×C173—41no—
98C2×Q8×C173C2311yescharacteristic
99C2×Q8×C173C2311yescharacteristic
100C2×D4×C173—21no—
101C2×D4×C173C2311yescharacteristic
102C22×C4×C173—21no—
103C22×C4×C173C2311yescharacteristic
104Q16×C173—41no—
105SD16×C173—22no—
106C2×C8×C173C2311yescharacteristic
107C2×C8×C173C2311yescharacteristic
108C4⋊C4×C173—21no—
109C4⋊C4×C173C2311yescharacteristic
110C22⋊C4×C173—21no—
111C22×D173—21no—
112C22×D173—42no—
113C22×D173—42no—
114C22×D173—81no—
115C22×D173C2×D411yescharacteristic
116Dic346—81no—
117D692—21no—
118D692—21no—
119C22⋊D173—22no—
120C22⋊D173—42no—
121C22⋊D173—81no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173—21no—
125C2×Dic173—81no—
126C2×Dic173—81no—
127C2×Dic173C2×D411yescharacteristic
128C2×Dic173C2×D411yescharacteristic
129C2×Dic173C2×D411yescharacteristic
130C4×D173—21no—
131C4×D173—21no—
132C4×D173—81no—
133#13063436—21no—
134#13063436—21no—
135C23×C173—21no—
136C23×C173C2×D411yescharacteristic
137Q8×C173—21no—
138Q8×C173—21no—
139Q8×C173—21no—
140Q8×C173—21no—
141D4×C173—21no—
142D4×C173—42no—
143D4×C173#4112yes—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173—21no—
149C2×C4×C173—42no—
150C2×C4×C173C2×D411yescharacteristic
151C2×C4×C173C2411yescharacteristic
152C8×C173—21no—
153C8×C173—21no—
154C22×C173—21no—
155C22×C173—22no—
156C22×C173—21no—
157C22×C173—21no—
158C22×C173—41no—
159C22×C173—41no—
160C22×C173C22×D411yescharacteristic
161C2×D173—21no—
162C2×D173—21no—
163C2×D173—41no—
164C2×D173—41no—
165C2×D173—41no—
166C2×D173—84no—
167C2×D173#13712yes—
168C4×C173—21no—
169C4×C173—41no—
170C4×C173—41no—
171C4×C173—41no—
172C4×C173C2×#4111yescharacteristic
173C4×C173H(5,𝔽2)11yescharacteristic
174Dic173—21no—
175Dic173—21no—
176Dic173—21no—
177Dic173—41no—
178Dic173—41no—
179Dic173—41no—
180C2×C173—42no—
181C2×C173—41no—
182C2×C173#54611yescharacteristic
183C2×C173C2×#13711yescharacteristic
184C2×C173#57611yescharacteristic
185D173—22no—
186D173—81no—
187C173#331411yescharacteristic, complemented, Sylow
188#3314—1731nomaximal, Sylow
189#554—1731no—
190#546—1731no—
191#489—1731no—
192#484—1731no—
193#480—1731no—
194#474—1731no—
195#468—1731no—
196#467—1731no—
197#464—1731no—
198#452D17311yescharacteristic, complemented
199#450—1731no—
200#440—1731no—
201#435—1731no—
202#417—1731no—
203C2×#102—1731no—
204C2×#41C2×D17311yescharacteristic
205C22×D4—1731no—
206C2×Q16C2×D17311yescharacteristic, complemented
207C2×SD16—1731no—
208C2×SD16C2×D17311yescharacteristic
209C2×M16—1731no—
210C22×C8—1731no—
211C4⋊Q8—1731no—
212#125—3461no—
213#124—1731no—
214#123—3461no—
215C22⋊Q8—3461no—
216C22⋊Q8—3461no—
217C22⋊Q8—1731no—
218C22⋊Q8—1731no—
219C22⋊Q8C2×D17311yescharacteristic, complemented
220#121—3461no—
221#121—1731no—
222#120—3461no—
223C4×Q8—1731no—
224C4×D4—3461no—
225C4×D4—1731no—
226C4×D4—1731no—
227C2×C4⋊C4—1731no—
228C2×C4⋊C4—1731no—
229C2×C22⋊C4—3461no—
230#107—1731no—
231#107—1731no—
232#107—1731no—
233C4⋊C8—1731no—
234#103—1731no—
235#103—1731no—
236#103—1731no—
237#103C2×D17311yescharacteristic, complemented
238#102—1731no—
239#102—1731no—
240#102—1731no—
241#102—1731no—
242#102—1731no—
243#102C2×D17311yescharacteristic
244C22⋊C8—1731no—
245C22⋊C8C2×D17311yescharacteristic, complemented
246#97—1731no—
247C24—3461no—
248#41—22no—
249#41—41no—
250C2×Q8—3461no—
251C2×Q8C22×D17311yescharacteristic
252C2×Q8C22×D17311yescharacteristic
253C2×D4—21no—
254C2×D4—3462no—
255C2×D4—3461no—
256C2×D4—3461no—
257C2×D4—1731no—
258C2×D4—1731no—
259C2×D4—6921no—
260C2×D4C22×D17311yescharacteristic
261C22×C4—21no—
262C22×C4—3461no—
263C22×C4—3461no—
264C22×C4—3461no—
265C22×C4—1731no—
266C22×C4C22×D17311yescharacteristic
267Q16—41no—
268SD16—22no—
269SD16—6921no—
270M16—6921no—
271C2×C8—1731no—
272C2×C8—1731no—
273C2×C8—6921no—
274C2×C8C22×D17311yescharacteristic
275C2×C8C22×D17311yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—3461no—
278C4⋊C4—3461no—
279C4⋊C4—3461no—
280C4⋊C4—3461no—
281C4⋊C4—3461no—
282C4⋊C4—3461no—
283C4⋊C4—1731no—
284C4⋊C4—1731no—
285C4⋊C4—1731no—
286C4⋊C4—1731no—
287C4⋊C4C22×D17311yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—3461no—
292C22⋊C4—3461no—
293C22⋊C4—3461no—
294C22⋊C4—3461no—
295C22⋊C4—1731no—
296C22⋊C4—1731no—
297C42—3461no—
298C42—1731no—
299C23—21no—
300C23—3461no—
301C23—13841no—
302C23—1731no—
303C23—6922no—
304C23—6922no—
305C23D4×D17311yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—21no—
309Q8—21no—
310Q8—13841no—
311D4—21no—
312D4—42no—
313D4—3462no—
314D4—3461no—
315D4—3461no—
316D4—13841no—
317D4—6922no—
318D4#1458382512yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—3461no—
326C2×C4—3461no—
327C2×C4—3461no—
328C2×C4—3461no—
329C2×C4—3461no—
330C2×C4—13841no—
331C2×C4—13841no—
332C2×C4—13841no—
333C2×C4—1731no—
334C2×C4—1731no—
335C2×C4—1731no—
336C2×C4D4×D17311yescharacteristic
337C2×C4C23×D17311yescharacteristic
338C8—21no—
339C8—21no—
340C8—3461no—
341C8—3461no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—22no—
346C22—41no—
347C22—41no—
348C22—3461no—
349C22—3461no—
350C22—13844no—
351C22—1732no—
352C22—6921no—
353C22—6921no—
354C22—6921no—
355C22C2×D4×D17311yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—3461no—
361C4—3461no—
362C4—3461no—
363C4—6921no—
364C4—6921no—
365C4—6921no—
366C4C2×#1458382511yescharacteristic
367C4#1914972411yescharacteristic
368C2—41no—
369C2—42no—
370C2—3462no—
371C2—13841no—
372C2#2601212811yescharacteristic
373C2#137×D17311yescharacteristic
374C2#2601229311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#452×C173C2
socleC22×C173C22×D4
prime radical#26012129C2
hypercenter#452D173
hypocenterC173#3314
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #26012199C2semidirect
3. #26011852C2semidirect
4. #26011825C2semidirect
5. #26011805C2semidirect
6. #26011769C2semidirect
7. #26011739C2semidirect
8. #26011729C2semidirect
9. #26011715C2semidirect
10. #26011640C2semidirect
11. #26011592C2semidirect
12. #26011565C2semidirect
13. #26011492C2semidirect
14. #102×D173C2semidirect
15. #452×C173C2semidirect
18. C2×#14583810C22semidirect
20. #19149678C22semidirect
31. Q8×Dic173C22semidirect
35. C2×C4⋊Dic173C22semidirect
38. #19149613C22semidirect
39. #19149613C22semidirect
40. #19149612C22semidirect
41. #19149608C22semidirect
42. #19149603C22semidirect
43. #19149603C22semidirect
44. #19149603C22semidirect
51. #19149589C22semidirect
53. C2×Q16×C173C22semidirect
55. C22⋊Q8×C173C22semidirect
56. #103×C173C22semidirect
58. C22⋊C8×C173C22semidirect
187. C173#3314semidirect
198. #452D173semidirect
206. C2×Q16C2×D173semidirect
219. C22⋊Q8C2×D173semidirect
237. #103C2×D173semidirect
245. C22⋊C8C2×D173semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#260121992
#260121292
#260118522
#260118252
#260118052
#260117692
#260117392
#260117292
#260117152
#260116402
#260115922
#260115652
#260114922
#102×D1732
#452×C1732
#3314173